Glossary of terms for Fermat's Last Theorem
Definitions for terms that are in boldface may be found elsewhere in the
glossary.
Note that some of these "definitions" are more or less in their exact
technical form, while others (due to the complexity of the concept) are
only intuitive descriptions.
- abelian group
- A group in which the operation is commutative.
- abelian variety
- An algebraic group that is also a complete algebraic
variety. The group is necesarily an abelian group.
- algebraic closure
- The smallest field that contains the roots of all polynomial
equations of one variable having coefficients lying in the field.
- algebraic curve
- An algebraic variety of dimension one. More informally,
it is the locus of points that satisfy a polynomial equation in
two variables.
- algebraic function
- A function whose dependent variable satisfies a polynomial
relationship with one or more independent variables.
- algebraic geometry
- The study of the geometric properties of the locus of points in
2 or more dimensions that satisfy sets of polynomial equations.
- algebraic group
- An algebraic variety that has a group structure where the
multiplication and inversion mappings are morphisms of
algebraic varieties. There are two distinct types of algebraic groups:
abelian varieties and linear algebraic groups.
- algebraic number
- Any solution of a polynomial equation of one variable whose
coeffients are in a specific field (usually the rational
numbers Q).
- algebraic variety
- One of the principal objects studied in algebraic geometry. A
generalization to higher dimensions of an algebraic curve.
It is, essentially, the locus of points that simultaneously satisfy
m polynomial equations in n variables, with m <. n.
- algebraically closed field
- A field that contains all solutions of 1-variable polynomial
equations with coefficients in the field.
- analytic continuation
- The process of extending an analytic function defined on
some domain (of the complex numbers) to a larger domain.
- analytic function
- A complex-valued function of a complex variable that is infinitely
differentiable. Equivalently, it is a complex function that can be
represented in the neighborhood of some point by a power series
about the point.
- automorphic function
- A meromorphic function of a complex variable that is
invariant under a group of transformations of the function's domain.
I. e., f is automorphic if f(Tz) = f(z) for every transformation
T in a given group.
- Birch and Swinnerton-Dyer conjecture
- The conjecture that the rank of the group of rational
points of an elliptic curve E is equal to the order of the zero
of the L-function L(E,s) of the curve at s=1.
- closed set
- The set complement of an open set in a topological
space. I. e., a set of all points of the space not in some open
set.
- complete algebraic variety
- An algebraic variety with the property that for any other
variety Y, the projection X x Y -> Y maps closed sets to
closed sets.
- conductor
- A integer associated with an elliptic curve that contains
information about the reduction of the curve at any prime
number.
- covering
- A mapping of toplogical spaces, p: X -> Y such that for any
y in Y there is an open set containing y whose pre-image (i. e.,
set of points that map to it) is a union of open sets of X.
- differential form
- A fundamental object related to the differential geometry of a
manifold. It is a way to define "partial differentiation" of
a function on a manifold in a manner that takes account of the
geometry of the manifold.
- differential geometry
- The study by techniques of differential calculus of the geometric
properties of manifolds.
- diophantine equation
- A polynomial equation with rational coefficients and one or more
variables for which integral or rational solutions are sought.
- Dirichlet series
- An infinite series of the form
. Typically
such a series converges in some part of the complex plane where Re(s)
is sufficiently large, and it defines an analytic function there.
- discriminant
- A function of the coefficients of a polynomial of one variable. It
is essentially the product of all possible differences of roots of the
polynomial. Therefore the discriminant is zero if and only if the
polynomial has repeated roots.
- eigen function
- A function in a vector space of functions whose image under some
linear transformation on the space is a constant multiple of itself.
I. e., there is some transformation T such that T(f) = cf for some
constant c.
- eigen value
- The constant multiple associated with a particular linear
transformation and eigen function. I. e. the constant c such
that T(f) = cf.
- elliptic curve
- A non-singular complete algebraic curve of genus one.
In more elementary terms, it is the locus of points satisfied by an
equation of the form ... where the right hand side of the equation has
no repeated roots.
- elliptic function
- A doubly-periodic meromorphic function. I. e., there are
two periods

and

, whose ratio isn't
a real number, such
that f(z+
) =
f(z+
) = f(z) for all complex z.
- equivalence relation
- A 2-place set-theoretic relation R that is reflexive, symmetric,
and transitive. I. e. xRx, xRy if and only if yRx, and xRy plus yRz
imply xRz. An equivalence relation on a set partitions the
set into disjoint equivalence classes (subsets of elements which are
all equivalent to each other).
- Euler product
- An infinite product of the form
for a
complex variable
s and complex values a
, with p a prime number.
When such a product
converges, it can be represented by a Dirichlet series. There
are generalizations whose factors are more complicated expressions
involving prime numbers.
- extension field
- A field that contains a smaller field. Usually it consists of
a given "base" field to which has been "adjoined" one or more roots of
a set of polynomial equations of one variable.
- field
- A mathematical system that has two distinct operations, where
both operations satisfy the axions of an abelian group.
Usually the operations are expressed as addition and multiplication.
(Zero is excluded from the multiplicative group since it has no
inverse.)
The most common examples of fields are the rational numbers Q,
the real
numbers R, the complex numbers C, and finite fields.
- finite field
- A field with a finite number of elements. The simplest
example is Z/pZ, also written F
,
the integers modulo a prime p.
All other finite fields are finite extensions of F
.
- Fourier series
- An infinite series of the form
.
Such series represent singly periodic meromorphic functions f(z),
where f(z+1) = f(z) for all z. There is an extensive theory developed
around the properties of such series, having many uses in both
theoretical and applied mathematics.
- fractional linear transformation
- A transformation of the complex plane of the form T(z) =
(az + b)/(cz + d) for a, b, c, d in C.
The set of all fractional linear transformations with coefficients in
Z forms a group called the modular group.
- functional equation
- An equation, which can be of many different forms, that prescribes
certain properties of a function. The solutions of the equation (if
any) are the functions having the property. Examples include
differential equations, the periodicity property f(z+t) = f(t),
the symmetry property of an automorphic function, and relationships
between function values at different points such as the functional
equations of the Riemann zeta function and L-functions.
- fundamental domain
- A connected region of the complex plane that contains exactly one
representative of each orbit under the action of some subgroup
of the modular group. For any given subgroup, there is usually
an obvious choice for the fundamental domain.
- Galois group
- A group of permutations of the roots of a polynomial equation of
one variable over some field.
- Galois representation
- A group representation of the Galois group of all
algebraic numbers over the rationals. Galois representations
can be constructed using any elliptic curve.
- Galois theory
- The theory of solutions of polynomial equations over a field.
The theory uses Galois groups to describe all possible
extension fields of a given field by means of a correspondence
with subgroups of the Galois group.
- general linear group
- The group of all invertible n-by-n matrices with coefficients in
a ring or, more usually, a field. Such a group is usually
denoted by GL
(R) or GL(n,R).
- genus
- A numerical integer invariant of an algebraic curve.
As applied to a topological object such as 2-dimensional manifold,
it can be interpreted as the number of "handles" the object has.
E. g. a sphere has genus 0, while a torus (donut shape) has genus 1.
There are various other definitions, such as the dimension of the
space of differential (1-)forms.
- group
- A mathematical system consisting of a set with an operation
between elements of the set and the properties that the operation
is associative (i. e. (ab)c = a(bc)), has an "identity element"
(i. e. 1a = a for all a), and all elements have inverses (i. e.
an a
with aa
= 1).
Groups are used pervasively in mathematics,
and they often express symmetry properties of other sets or objects.
- group representation
- A homomorphism from an abstract group to a
general linear group. In general, it need not be either
injective (1-to-1) or surjective (onto). Group representations
have numerous theoretical and applied uses, since matrix groups
have well-known properties and are easy to compute with.
- holomorphic function
- An analytic function. The terms are used interchangably.
- homeomorphism
- A 1-to-1 mapping between topological spaces that preserves
the topological structure, i. e. the open sets. Homeomorphic spaces
have essentially the same topological properties.
- homomorphism
- A mapping between mathematical structures of the same type
(e. g. groups or rings) that preserves the structure,
i. e. f(ab) = f(a)f(b).
- isomorphism
- A mapping between mathematical strucures of the same type that
preserves the structure and is both injective (1-to-1) and surjective
(onto). Isomorphic objects are essentially the same with respect to
the preserved structure.
- kernel
- A subset of the domain of a homomorphism consisting of all
elements that map to the identity element. The kernel is always a
sub-object of appropriate type. E. g. the kernel of a group
homomorphism is a subgroup.
- lattice
- A discrete subgroup of the additive group of complex
numbers. Concretely, it is the set of all complex numbers of the
form n


+ n

for integers n
, n
and
"periods" 
and 
(whose ratio is
not a real number).
- L-function
- A complex function which can usually be represented as a
Dirichlet series or Euler product and which
expresses arithmetic properties of some mathematical construct
such as an elliptic curve or modular function.
L-functions are a powerful theoretical tool for "encoding" arithmetic
information in a single object.
- Lie group
- An analytic manifold G that has a group structure
such that that the map (x,y) -> xy
from G x G to
G is analytic
(i. e. infinitely differentiable). The general linear groups
GL
(R), GL
(C),
and their subgroups, are the most
common examples. There is an extensive and deep theory of Lie groups,
with many theoretical and applied uses.
- linear algebraic group
- A group that is isomorphic to a subgroup of a general linear
group. It is one of the two distinct types of algebraic groups.
- manifold
- A topological space that is "locally Euclidean" in the sense that
every point is contained in an open set that is homeomorphic
to an open set of Euclidean n-space R
.
Additional conditions such as
differentiability are often imposed. The manifold is the primary
object of study in differential geomentry.
- meromorphic function
- An function that is an analytic function except at a
discrete set of points where it has singularity called "poles".
At such a point, the power series expansion of the function has a
finite number of terms with negative powers of z.
- modular curve
- A complete algebraic variety which is an algebraic curve
that is essentially the quotient space of the upper half of the
complex plane by the action of a subgroup of finite index of the
modular group. This space is "compactified" by the addition of
a finite number of points in the same way as the Riemann sphere
is constructed.
- modular elliptic curve
- An elliptic curve E for which there is a modular curve X
of a certain kind and a surjective map X -> E. Such an elliptic curve
is said to have a "parameterization by modular functions". There are
equivalent definitions, the simplest of which is that there exists a
modular form whose L-function is the same as that of E.
The Taniyama-Shimura conjecture states that every elliptic
curve is modular.
- modular form
- A holomorphic modular function. The term is usually applied in
a more general sense in the same way as with modular functions, i. e.
including functions with non-zero weight and with respect to subgroups
of finite index in the modular group.
- modular function
- A special type of automorphic function where the group
involved is the modular group. The term is usually applied in a
more general way. First, the automorphicity condition is relaxed to
f((az+b)/(cz+d)) = (cz+d)
f(z), where the
integer k is called the
"weight" of f. Second, the condition may be applied only for certain
subgroups of finite index in the modular group.
- modular group
- The group of all fractional linear transformations of the
complex plane with coefficients in Z.
This is essentially the same, up to a factor of
1,
as SL
(Z), the group of
2-by-2 matrices with entries in Z and determinant 1, so
sometimes SL
(Z) is referred to as the
modular group.
- morphism
- A mapping between two mathematical objects of the same type,
such as topological spaces or groups, that preserves
the essential structure of the object.
- neighborhood
- An open set in a topological space that contains
a specific point.
- normal subgroup
- A subgroup H of a group G with the property that H =
xHx
for any x in G. The kernel of a
group homomorphism
is always normal, and a normal subgroup is the kernel of the
projection map G -> G/H onto the quotient group.
- open set
- A member of a class of subsets of a topological space
that satisfy certain axioms and define the topology.
- orbit
- A set of points of the complex plane containing all points which are
equivalent under all transformations in some subgroup of the
modular group.
- p-adic numbers
- A field containing the ordinary rational numbers, defined
for any prime p. There are several ways the construction can be
performed, such as formal power series in p, as the "completion"
of Q with respect to a metric based on divisibility by p,
or as the field of quotients of a "projective limit" of the
sequence of groups Z/p
Z.
There are p-adic
analogues of many concepts of complex analysis.
- projective plane
- A geometric construct frequently used in algebraic geometry
to make the equations easier to deal with and avoid having to treat
the "point at infinity" as a special case.
- quotient group
- The group obtained from a group G with normal subgroup
H by putting a natural group structure on the set of equivalence
classes of elements of G under the equivalence relation that
x
y if and only xy
H.
The quotient group is denoted by G/H.
- quotient space
- A topological space obtained in a natural way from another
space on which there is an equivalence relation. The points of
the quotient space are equivalence classes, and the topology is the
strongest one such that the projection map is continuous.
- rank
- A numerical invariant of a finitely generated abelian group.
Such a group is isomorphic to a product of a finte group and a
finite number of infinite cyclic groups. The rank is the number of
infinite cyclic groups in this product.
- reduction
- The process of viewing an algebraic curve defined by a
polynomial with integral coefficients as a curve over the finite
field F
for some prime p.
- Riemann sphere
- The Riemann surface obtained by "compactifying" the
complex plane by adding a "point at infinity" and appropriate
neighborhoods of that point.
- Riemann surface
- A topological manifold that serves as the
domain of definition of a single-valued algebraic function.
The precise construction is rather technical, but greatly simplifies
many ideas in complex function theory.
- Riemann zeta function
- The Dirichlet series
. The zeta
function has many surprising
analytic and number theoretic properties, and has been one of the
central objects of study in analytic number theory. Many of its
properties can be generalized to L-functions.
- ring
- A mathematical system that has two operations, usually called
addition and multiplication. A ring is an abelian group with
respect to addition. Multiplication is associative and distributive
with respect to addition.
- special linear group
- A subgroup of the general linear group consisting of matrices
having determinant 1. It is usually denoted SL
(R)
or SL(n,R) for some ring R.
- Taniyama-Shimura conjecture
- The conjecture that every elliptic curve is actually a
modular elliptic curve. It has now been proven for the case
where the conductor of the curve is square-free (the
"semistable" case).
- topological space
- A set together with a collection of subsets, called open sets
that satisfy certain axioms. The open sets endow the space with a
concept of "nearness" between any two points. This is a generalization
of the concept of nearness obtained from a numerical measure of "distance"
between two points.
- vector space
- A mathematical system consisting of a set of points ("vectors")
that form an abelian group and which allow for "multiplication"
by elements of some "field". Examples of vector spaces include
n-tuples of elements of a field and
various classes of analytic functions. A vector space is the
fundamental object of study in linear algebra.
Back to Fermat's Last Theorem Home Page
Copyright © 1996 by Charles Daney, All Rights Reserved
Last updated: April 4, 1996